3.2.10 \(\int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{(1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x) \sqrt {a-b x^3}} \, dx\) [110]

Optimal. Leaf size=75 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {a} \left (1-\sqrt [3]{\frac {b}{a}} x\right )}{\sqrt {a-b x^3}}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt {a} \sqrt [3]{\frac {b}{a}}} \]

[Out]

2*arctanh((1-(b/a)^(1/3)*x)*a^(1/2)*(-3+2*3^(1/2))^(1/2)/(-b*x^3+a)^(1/2))/(b/a)^(1/3)/a^(1/2)/(-3+2*3^(1/2))^
(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2165, 212} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt {a} \left (1-x \sqrt [3]{\frac {b}{a}}\right )}{\sqrt {a-b x^3}}\right )}{\sqrt {2 \sqrt {3}-3} \sqrt {a} \sqrt [3]{\frac {b}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[3] - (b/a)^(1/3)*x)/((1 - Sqrt[3] - (b/a)^(1/3)*x)*Sqrt[a - b*x^3]),x]

[Out]

(2*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[a]*(1 - (b/a)^(1/3)*x))/Sqrt[a - b*x^3]])/(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[a]*
(b/a)^(1/3))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2165

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{k = Simplify[(d*e
+ 2*c*f)/(c*f)]}, Dist[(1 + k)*(e/d), Subst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + (1 + k)*d*(x/c))/Sqrt[a +
b*x^3]], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6
, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 0]

Rubi steps

\begin {align*} \int \frac {1+\sqrt {3}-\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}-\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a-b x^3}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{1+\left (3-2 \sqrt {3}\right ) a x^2} \, dx,x,\frac {1-\sqrt [3]{\frac {b}{a}} x}{\sqrt {a-b x^3}}\right )}{\sqrt [3]{\frac {b}{a}}}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {a} \left (1-\sqrt [3]{\frac {b}{a}} x\right )}{\sqrt {a-b x^3}}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt {a} \sqrt [3]{\frac {b}{a}}}\\ \end {align*}

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Mathematica [A]
time = 7.35, size = 84, normalized size = 1.12 \begin {gather*} -\frac {2 \sqrt [6]{\frac {b}{a}} \tanh ^{-1}\left (\frac {\sqrt [6]{\frac {b}{a}} \sqrt {b+\frac {2 b}{\sqrt {3}}} \sqrt {a-b x^3}}{-a \left (\frac {b}{a}\right )^{2/3}+b x}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sqrt[3] - (b/a)^(1/3)*x)/((1 - Sqrt[3] - (b/a)^(1/3)*x)*Sqrt[a - b*x^3]),x]

[Out]

(-2*(b/a)^(1/6)*ArcTanh[((b/a)^(1/6)*Sqrt[b + (2*b)/Sqrt[3]]*Sqrt[a - b*x^3])/(-(a*(b/a)^(2/3)) + b*x)])/(Sqrt
[-3 + 2*Sqrt[3]]*Sqrt[b])

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\sqrt {3}}{\left (1-\left (\frac {b}{a}\right )^{\frac {1}{3}} x -\sqrt {3}\right ) \sqrt {-b \,x^{3}+a}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-(b/a)^(1/3)*x+3^(1/2))/(1-(b/a)^(1/3)*x-3^(1/2))/(-b*x^3+a)^(1/2),x)

[Out]

int((1-(b/a)^(1/3)*x+3^(1/2))/(1-(b/a)^(1/3)*x-3^(1/2))/(-b*x^3+a)^(1/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-(b/a)^(1/3)*x+3^(1/2))/(1-(b/a)^(1/3)*x-3^(1/2))/(-b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((x*(b/a)^(1/3) - sqrt(3) - 1)/(sqrt(-b*x^3 + a)*(x*(b/a)^(1/3) + sqrt(3) - 1)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (57) = 114\).
time = 0.75, size = 1330, normalized size = 17.73 \begin {gather*} \left [\frac {1}{2} \, \sqrt {\frac {1}{3}} \sqrt {\frac {{\left (2 \, \sqrt {3} + 3\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}}{b}} \log \left (\frac {b^{8} x^{24} + 1840 \, a b^{7} x^{21} + 67264 \, a^{2} b^{6} x^{18} + 58624 \, a^{3} b^{5} x^{15} + 504064 \, a^{4} b^{4} x^{12} - 2140160 \, a^{5} b^{3} x^{9} + 3100672 \, a^{6} b^{2} x^{6} - 1089536 \, a^{7} b x^{3} + 28672 \, a^{8} - 4 \, \sqrt {\frac {1}{3}} {\left ({\left (3 \, a b^{7} x^{22} + 2688 \, a^{2} b^{6} x^{19} + 56952 \, a^{3} b^{5} x^{16} + 93504 \, a^{4} b^{4} x^{13} - 63552 \, a^{5} b^{3} x^{10} + 377856 \, a^{6} b^{2} x^{7} - 314880 \, a^{7} b x^{4} + 24576 \, a^{8} x - 2 \, \sqrt {3} {\left (a b^{7} x^{22} + 764 \, a^{2} b^{6} x^{19} + 16860 \, a^{3} b^{5} x^{16} + 19792 \, a^{4} b^{4} x^{13} + 42368 \, a^{5} b^{3} x^{10} - 104448 \, a^{6} b^{2} x^{7} + 90880 \, a^{7} b x^{4} - 7168 \, a^{8} x\right )}\right )} \sqrt {-b x^{3} + a} \left (\frac {b}{a}\right )^{\frac {2}{3}} + 2 \, {\left (30 \, a b^{7} x^{21} + 5010 \, a^{2} b^{6} x^{18} + 44640 \, a^{3} b^{5} x^{15} + 21360 \, a^{4} b^{4} x^{12} + 79872 \, a^{5} b^{3} x^{9} - 233856 \, a^{6} b^{2} x^{6} + 86016 \, a^{7} b x^{3} - 3072 \, a^{8} - \sqrt {3} {\left (17 \, a b^{7} x^{21} + 2920 \, a^{2} b^{6} x^{18} + 24864 \, a^{3} b^{5} x^{15} + 26576 \, a^{4} b^{4} x^{12} - 56000 \, a^{5} b^{3} x^{9} + 115968 \, a^{6} b^{2} x^{6} - 56320 \, a^{7} b x^{3} + 1024 \, a^{8}\right )}\right )} \sqrt {-b x^{3} + a} \left (\frac {b}{a}\right )^{\frac {1}{3}} + 6 \, {\left (81 \, a b^{7} x^{20} + 4752 \, a^{2} b^{6} x^{17} + 14472 \, a^{3} b^{5} x^{14} + 24192 \, a^{4} b^{4} x^{11} - 39744 \, a^{5} b^{3} x^{8} + 69120 \, a^{6} b^{2} x^{5} - 13824 \, a^{7} b x^{2} - \sqrt {3} {\left (47 \, a b^{7} x^{20} + 2724 \, a^{2} b^{6} x^{17} + 8976 \, a^{3} b^{5} x^{14} + 4928 \, a^{4} b^{4} x^{11} + 32448 \, a^{5} b^{3} x^{8} - 37632 \, a^{6} b^{2} x^{5} + 8192 \, a^{7} b x^{2}\right )}\right )} \sqrt {-b x^{3} + a}\right )} \sqrt {\frac {{\left (2 \, \sqrt {3} + 3\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}}{b}} + 8 \, {\left (3 \, a b^{7} x^{23} + 1077 \, a^{2} b^{6} x^{20} + 13320 \, a^{3} b^{5} x^{17} + 19200 \, a^{4} b^{4} x^{14} - 111360 \, a^{5} b^{3} x^{11} + 345024 \, a^{6} b^{2} x^{8} - 328704 \, a^{7} b x^{5} + 61440 \, a^{8} x^{2} - 2 \, \sqrt {3} {\left (a b^{7} x^{23} + 299 \, a^{2} b^{6} x^{20} + 4260 \, a^{3} b^{5} x^{17} - 1520 \, a^{4} b^{4} x^{14} + 26720 \, a^{5} b^{3} x^{11} - 105024 \, a^{6} b^{2} x^{8} + 93184 \, a^{7} b x^{5} - 17920 \, a^{8} x^{2}\right )}\right )} \left (\frac {b}{a}\right )^{\frac {2}{3}} - 32 \, \sqrt {3} {\left (35 \, a b^{7} x^{21} + 1141 \, a^{2} b^{6} x^{18} + 2544 \, a^{3} b^{5} x^{15} - 6760 \, a^{4} b^{4} x^{12} + 39520 \, a^{5} b^{3} x^{9} - 55680 \, a^{6} b^{2} x^{6} + 19712 \, a^{7} b x^{3} - 512 \, a^{8}\right )} + 32 \, {\left (9 \, a b^{7} x^{22} + 846 \, a^{2} b^{6} x^{19} + 4617 \, a^{3} b^{5} x^{16} - 5472 \, a^{4} b^{4} x^{13} + 43776 \, a^{5} b^{3} x^{10} - 98496 \, a^{6} b^{2} x^{7} + 59328 \, a^{7} b x^{4} - 4608 \, a^{8} x - \sqrt {3} {\left (5 \, a b^{7} x^{22} + 505 \, a^{2} b^{6} x^{19} + 2130 \, a^{3} b^{5} x^{16} + 4928 \, a^{4} b^{4} x^{13} - 28688 \, a^{5} b^{3} x^{10} + 53760 \, a^{6} b^{2} x^{7} - 35200 \, a^{7} b x^{4} + 2560 \, a^{8} x\right )}\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}}{b^{8} x^{24} - 80 \, a b^{7} x^{21} + 2368 \, a^{2} b^{6} x^{18} - 30080 \, a^{3} b^{5} x^{15} + 121984 \, a^{4} b^{4} x^{12} + 240640 \, a^{5} b^{3} x^{9} + 151552 \, a^{6} b^{2} x^{6} + 40960 \, a^{7} b x^{3} + 4096 \, a^{8}}\right ), -\sqrt {\frac {1}{3}} \sqrt {-\frac {{\left (2 \, \sqrt {3} + 3\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\sqrt {-b x^{3} + a} b x^{2} - 2 \, \sqrt {-b x^{3} + a} {\left (\sqrt {3} a x - 2 \, a x\right )} \left (\frac {b}{a}\right )^{\frac {2}{3}} + 2 \, \sqrt {-b x^{3} + a} {\left (\sqrt {3} a - a\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {{\left (2 \, \sqrt {3} + 3\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}}}{b}}}{2 \, {\left (b x^{3} - a\right )}}\right )\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-(b/a)^(1/3)*x+3^(1/2))/(1-(b/a)^(1/3)*x-3^(1/2))/(-b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

[1/2*sqrt(1/3)*sqrt((2*sqrt(3) + 3)*(b/a)^(1/3)/b)*log((b^8*x^24 + 1840*a*b^7*x^21 + 67264*a^2*b^6*x^18 + 5862
4*a^3*b^5*x^15 + 504064*a^4*b^4*x^12 - 2140160*a^5*b^3*x^9 + 3100672*a^6*b^2*x^6 - 1089536*a^7*b*x^3 + 28672*a
^8 - 4*sqrt(1/3)*((3*a*b^7*x^22 + 2688*a^2*b^6*x^19 + 56952*a^3*b^5*x^16 + 93504*a^4*b^4*x^13 - 63552*a^5*b^3*
x^10 + 377856*a^6*b^2*x^7 - 314880*a^7*b*x^4 + 24576*a^8*x - 2*sqrt(3)*(a*b^7*x^22 + 764*a^2*b^6*x^19 + 16860*
a^3*b^5*x^16 + 19792*a^4*b^4*x^13 + 42368*a^5*b^3*x^10 - 104448*a^6*b^2*x^7 + 90880*a^7*b*x^4 - 7168*a^8*x))*s
qrt(-b*x^3 + a)*(b/a)^(2/3) + 2*(30*a*b^7*x^21 + 5010*a^2*b^6*x^18 + 44640*a^3*b^5*x^15 + 21360*a^4*b^4*x^12 +
79872*a^5*b^3*x^9 - 233856*a^6*b^2*x^6 + 86016*a^7*b*x^3 - 3072*a^8 - sqrt(3)*(17*a*b^7*x^21 + 2920*a^2*b^6*x
^18 + 24864*a^3*b^5*x^15 + 26576*a^4*b^4*x^12 - 56000*a^5*b^3*x^9 + 115968*a^6*b^2*x^6 - 56320*a^7*b*x^3 + 102
4*a^8))*sqrt(-b*x^3 + a)*(b/a)^(1/3) + 6*(81*a*b^7*x^20 + 4752*a^2*b^6*x^17 + 14472*a^3*b^5*x^14 + 24192*a^4*b
^4*x^11 - 39744*a^5*b^3*x^8 + 69120*a^6*b^2*x^5 - 13824*a^7*b*x^2 - sqrt(3)*(47*a*b^7*x^20 + 2724*a^2*b^6*x^17
+ 8976*a^3*b^5*x^14 + 4928*a^4*b^4*x^11 + 32448*a^5*b^3*x^8 - 37632*a^6*b^2*x^5 + 8192*a^7*b*x^2))*sqrt(-b*x^
3 + a))*sqrt((2*sqrt(3) + 3)*(b/a)^(1/3)/b) + 8*(3*a*b^7*x^23 + 1077*a^2*b^6*x^20 + 13320*a^3*b^5*x^17 + 19200
*a^4*b^4*x^14 - 111360*a^5*b^3*x^11 + 345024*a^6*b^2*x^8 - 328704*a^7*b*x^5 + 61440*a^8*x^2 - 2*sqrt(3)*(a*b^7
*x^23 + 299*a^2*b^6*x^20 + 4260*a^3*b^5*x^17 - 1520*a^4*b^4*x^14 + 26720*a^5*b^3*x^11 - 105024*a^6*b^2*x^8 + 9
3184*a^7*b*x^5 - 17920*a^8*x^2))*(b/a)^(2/3) - 32*sqrt(3)*(35*a*b^7*x^21 + 1141*a^2*b^6*x^18 + 2544*a^3*b^5*x^
15 - 6760*a^4*b^4*x^12 + 39520*a^5*b^3*x^9 - 55680*a^6*b^2*x^6 + 19712*a^7*b*x^3 - 512*a^8) + 32*(9*a*b^7*x^22
+ 846*a^2*b^6*x^19 + 4617*a^3*b^5*x^16 - 5472*a^4*b^4*x^13 + 43776*a^5*b^3*x^10 - 98496*a^6*b^2*x^7 + 59328*a
^7*b*x^4 - 4608*a^8*x - sqrt(3)*(5*a*b^7*x^22 + 505*a^2*b^6*x^19 + 2130*a^3*b^5*x^16 + 4928*a^4*b^4*x^13 - 286
88*a^5*b^3*x^10 + 53760*a^6*b^2*x^7 - 35200*a^7*b*x^4 + 2560*a^8*x))*(b/a)^(1/3))/(b^8*x^24 - 80*a*b^7*x^21 +
2368*a^2*b^6*x^18 - 30080*a^3*b^5*x^15 + 121984*a^4*b^4*x^12 + 240640*a^5*b^3*x^9 + 151552*a^6*b^2*x^6 + 40960
*a^7*b*x^3 + 4096*a^8)), -sqrt(1/3)*sqrt(-(2*sqrt(3) + 3)*(b/a)^(1/3)/b)*arctan(-1/2*sqrt(1/3)*(sqrt(-b*x^3 +
a)*b*x^2 - 2*sqrt(-b*x^3 + a)*(sqrt(3)*a*x - 2*a*x)*(b/a)^(2/3) + 2*sqrt(-b*x^3 + a)*(sqrt(3)*a - a)*(b/a)^(1/
3))*sqrt(-(2*sqrt(3) + 3)*(b/a)^(1/3)/b)/(b*x^3 - a))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \sqrt [3]{\frac {b}{a}} - \sqrt {3} - 1}{\sqrt {a - b x^{3}} \left (x \sqrt [3]{\frac {b}{a}} - 1 + \sqrt {3}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-(b/a)**(1/3)*x+3**(1/2))/(1-(b/a)**(1/3)*x-3**(1/2))/(-b*x**3+a)**(1/2),x)

[Out]

Integral((x*(b/a)**(1/3) - sqrt(3) - 1)/(sqrt(a - b*x**3)*(x*(b/a)**(1/3) - 1 + sqrt(3))), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-(b/a)^(1/3)*x+3^(1/2))/(1-(b/a)^(1/3)*x-3^(1/2))/(-b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(c
onst gen &

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\sqrt {3}-x\,{\left (\frac {b}{a}\right )}^{1/3}+1}{\sqrt {a-b\,x^3}\,\left (\sqrt {3}+x\,{\left (\frac {b}{a}\right )}^{1/3}-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3^(1/2) - x*(b/a)^(1/3) + 1)/((a - b*x^3)^(1/2)*(3^(1/2) + x*(b/a)^(1/3) - 1)),x)

[Out]

int(-(3^(1/2) - x*(b/a)^(1/3) + 1)/((a - b*x^3)^(1/2)*(3^(1/2) + x*(b/a)^(1/3) - 1)), x)

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